Question

which series of transformations occurred to move the l shape in quadrant 1 to the l shape in quadrant 3? a 90 - degree rotation counterclockwise about the origin and then a reflection a translation and then a reflection a reflection and then a reflection a reflection and then a 180 - degree rotation about the origin

Explanation:

Step1: Analyze 180° rotation about origin

A 180° rotation about the origin \((x,y)\to(-x,-y)\). Let's take a vertex of the red L - shape (quadrant 1, say \((2, - 3)\) approximately). After 180° rotation, it becomes \((-2,3)\), which is in quadrant 2. But the blue shape is in quadrant 3? Wait, no, let's re - examine the coordinates. Wait, the red shape is in quadrant 1? Wait, no, the grid: x - axis and y - axis. Wait, the red shape is in the region where x is negative (left of y - axis) and y is negative (below x - axis)? Wait, no, the axes: the x - axis goes from left (negative) to right (positive), y - axis from bottom (negative) to top (positive). Wait, the red shape: let's take a point, say the top - right corner of red: \((-2,-4)\)? Wait, no, maybe I got the axes reversed. Wait, the y - axis is vertical, x - axis horizontal. Let's assume the standard coordinate system: right is positive x, up is positive y. So the red shape is in the third quadrant (x negative, y negative), blue in the first quadrant? Wait, no, the problem says "move the L - shape in quadrant 1 to the L - shape in quadrant 3"? Wait, maybe I misread. Wait, the problem: "Which series of transformations occurred to move the L - shape in quadrant 1 to the L - shape in quadrant 3?" Wait, looking at the graph: red is in, say, quadrant 2 (x negative, y positive)? No, the red shape: x from - 4 to - 2, y from - 1 to - 4? Wait, no, the grid lines: each square is 1 unit. Let's take a key point of the red L - shape: let's say the corner at ( - 3, - 1) (x=-3, y = - 1). The blue L - shape: a corner at (3,1). Wait, no, maybe the correct approach is to check the transformation. A 180° rotation about the origin: \((x,y)\to(-x,-y)\). If the original is in quadrant 1 (\(x>0,y>0\)), after 180° rotation, it's in quadrant 3 (\(x<0,y<0\))? No, wait, quadrant 1: \(x>0,y>0\); quadrant 3: \(x<0,y<0\). Wait, but the options: "a reflection and then a 180 - degree rotation about the origin" – no, let's check the option "a reflection and then a 180 - degree rotation about the origin" or "a 180 - degree rotation about the origin" or "a reflection and then a reflection"? Wait, no, the option "a reflection and then a 180 - degree rotation about the origin" – no, let's think about the orientation. The L - shape: the red and blue have opposite orientations. A 180° rotation about the origin will reverse both x and y signs and also rotate the shape 180°, which can match the orientation. Wait, another approach: a 180° rotation about the origin is equivalent to two reflections (over x - axis and y - axis, or over y = x and y=-x, but more simply, (x,y)\to(-x,-y)). If the original is in quadrant 1 (\(x>0,y>0\)), after 180° rotation, it's in quadrant 3 (\(x<0,y<0\)). But also, a reflection and then 180°? Wait, no, the option "a reflection and then a 180 - degree rotation about the origin" – no, let's check the options again. Wait, the options:

1. "a 90 - degree counterclockwise rotation about the origin and then a reflection"
2. "a translation and then a reflection"
3. "a reflection and then a reflection"
4. "a reflection and then a 180 - degree rotation about the origin"

Wait, no, the user's image has four options:

- Option 1: a 90 - degree counterclockwise rotation about the origin and then a reflection
- Option 2: a translation and then a reflection
- Option 3: a reflection and then a reflection
- Option 4: a reflection and then a 180 - degree rotation about the origin

Wait, no, maybe the correct transformation is a 180° rotation about the origin, but among the options, "a reflection and then a 180 - degree rotation about the origin" – no, wait, let's take a point. Suppose the original L - shape in quadrant 1 has a point (2,3). After a 180° rotation about the origin, it becomes (- 2,-3), which is in quadrant 3. But if we first reflect over, say, the x - axis: (2,3)\to(2,-3) (quadrant 4), then 180° rotation: (2,-3)\to(-2,3) (quadrant 2). No. Wait, maybe the original is in quadrant 2, and we need to get to quadrant 3. Wait, I think I made a mistake in quadrant identification. Let's use the standard:

Quadrant 1: \(x>0,y>0\)

Quadrant 2: \(x<0,y>0\)

Quadrant 3: \(x<0,y<0\)

Quadrant 4: \(x>0,y<0\)

Looking at the red shape: x - coordinates negative (left of y - axis), y - coordinates negative (below x - axis) – so quadrant 3. Blue shape: x - coordinates positive (right of y - axis), y - coordinates positive (above x - axis) – quadrant 1. Wait, the problem says "move the L - shape in quadrant 1 to the L - shape in quadrant 3". So original (quadrant 1: \(x>0,y>0\)) to target (quadrant 3: \(x<0,y<0\)). A 180° rotation about the origin takes \((x,y)\) to \((-x,-y)\), so (2,3)\to(-2,-3) (quadrant 3). But the options: one of the options is "a reflection and then a 180 - degree rotation about the origin"? No, wait, the option "a reflection and then a 180 - degree rotation about the origin" – no, let's check the other option: "a 180 - degree rotation about the origin" – but the options given:

Wait, the four options (from the image, as per the user's input):

1. a 90 - degree counterclockwise rotation about the origin and then a reflection

2. a translation and then a reflection

3. a reflection and then a reflection

4. a reflection and then a 180 - degree rotation about the origin

Wait, no, maybe the correct answer is "a reflection and then a 180 - degree rotation about the origin" – no, wait, let's think about the orientation. The L - shape: the red and blue have opposite "handedness" (mirror image) and also rotated. Wait, a 180° rotation plus a reflection? No, the key is that a 180° rotation about the origin combined with a reflection? Wait, no, let's take a simple case. Suppose the original point is (1,1) (quadrant 1). After a 180° rotation, it's (- 1,-1) (quadrant 3). But if we first reflect over the y - axis: (1,1)\to(-1,1) (quadrant 2), then 180° rotation: (-1,1)\to(1,-1) (quadrant 4). No. Wait, maybe the original is in quadrant 1, and we need to get to quadrant 3. The transformation that takes (x,y) to (-x,-y) is 180° rotation about the origin. But the options: the option "a reflection and then a 180 - degree rotation about the origin" – no, maybe the correct option is "a reflection and then a 180 - degree rotation about the origin" is not, but "a 180 - degree rotation about the origin" – but the options given have "a reflection and then a 180 - degree rotation about the origin" as one of them? Wait, no, the user's options (from the image text):

Wait, the four cards:

1. "a 90 - degree counterclockwise rotation about the origin and then a reflection"

2. "a translation and then a reflection"

3. "a reflection and then a reflection"

4. "a reflection and then a 180 - degree rotation about the origin"

Wait, I think I made a mistake. Let's look at the orientation of the L - shape. The red L - shape and blue L - shape: if we reflect the blue (quadrant 1) over the x - axis, we get to quadrant 4, then reflect over the y - axis, we get to quadrant 3. But that's two reflections, which is equivalent to a 180° rotation. Wait, two reflections (over x and y axes) is a 180° rotation. But the option "a reflection and then a reflection" – that would be two reflections, which is a 180° rotation. Wait, no, two reflections (over perpendicular axes, like x and y) is a 180° rotation. So if we reflect the L - shape in quadrant 1 over the x - axis (to quadrant 4) and then over the y - axis (to quadrant 3), that's two reflections, which is the same as a 180° rotation. But the option "a reflection and then a reflection" – yes, that would move from quadrant 1 to quadrant 3. Wait, no, first reflection (say over x - axis): (x,y)\to(x,-y) (quadrant 4), second reflection (over y - axis): (x,-y)\to(-x,-y) (quadrant 3). So two reflections. But the option "a reflection and then a reflection" – that's the third option? Wait, the user's options:

Looking at the text on the cards:

- Top - left: "a reflection and then a reflection"

- Top - right: "a 90 - degree counterclockwise rotation about the origin and then a reflection"

- Bottom - left: "a reflection and then a 180 - degree rotation about the origin"

- Bottom - right: "a translation and then a reflection"

Wait, I think I misread earlier. So the correct transformation: to go from quadrant 1 (x>0,y>0) to quadrant 3 (x<0,y<0), we can do two reflections (over x - axis and y - axis), which is "a reflection and then a reflection". Alternatively, a 180° rotation. But among the options, "a reflection and then a reflection" is an option. Wait, but let's check the orientation. The L - shape: if we reflect over x - axis (flips vertically), then reflect over y - axis (flips horizontally), the result is a 180° rotation, which matches the orientation of the target. So the correct option is "a reflection and then a reflection"? Wait, no, the option "a reflection and then a 180 - degree rotation about the origin" – no, two reflections are equivalent to a 180° rotation. Wait, maybe the correct answer is "a reflection and then a 180 - degree rotation about the origin" is wrong, and "a reflection and then a reflection" is correct. But wait, let's take a point (2,3) in quadrant 1.

- After first reflection (over x - axis): (2,-3) (quadrant 4)

- After second reflection (over y - axis): (-2,-3) (quadrant 3)

Yes, that works. So the transformation is two reflections, i.e., "a reflection and then a reflection". But wait, the option "a reflection and then a 180 - degree rotation about the origin" – let's see: (2,3) reflect over x - axis: (2,-3), then 180° rotation: (-2,3) (quadrant 2) – no, that's not quadrant 3. So the correct option is "a reflection and then a reflection".

Wait, but the problem says "move the L - shape in quadrant 1 to the L - shape in quadrant 3". So the correct transformation is two reflections, which is "a reflection and then a reflection".

Step2: Eliminate other options

- Option "a 90 - degree counterclockwise rotation about the origin and then a reflection": A 90° counterclockwise rotation of (x,y) is (-y,x). If original is (2,3), after 90° CCW rotation: (-3,2) (quadrant 2), then reflection (say over x - axis): (-3,-2) (quadrant 3). But the orientation of the L - shape would be different from the target. So this is incorrect.

- Option "a translation and then a reflection": Translation moves the shape without rotating or reflecting its orientation, then reflection. But to get from quadrant 1 to 3, translation can't change the quadrant (since translation is (x + h,y + k)), so this is incorrect.

- Option "a reflection and then a 180 - degree rotation about the origin": As we saw earlier, (2,3) reflect over x - axis: (2,-3), then 180° rotation: (-2,3) (quadrant 2), not 3. So incorrect.

- Option "a reflection and then a reflection": As shown, (2,3) reflect over x - axis: (2,-3), reflect over y - axis: (-2,-3) (quadrant 3), which matches. So this is correct.

Answer:

The option "a reflection and then a reflection" (the top - left card, assuming the cards are arranged as top - left: "a reflection and then a reflection", top - right: "a 90 - degree counterclockwise rotation about the origin and then a reflection", bottom - left: "a reflection and then a 180 - degree rotation about the origin", bottom - right: "a translation and then a reflection")